Local Search Yields Approximation Schemes for $k$-Means and $k$-Median in Euclidean and Minor-Free Metrics

Abstract

We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs; (2) $k$-median and $k$-means in edge-weighted planar graphs; and (3) $k$-means in Euclidean space of bounded dimension. Our first and second results extend to minor-closed families of graphs. All our results extend to cost functions that are the $p$th power of the shortest-path distance. The algorithm is local search, where the local neighborhood of a solution $S$ consists of all solutions obtained from $S$ by removing and adding $1/\varepsilon^{\Theta(1)}$ centers.